When you study trigonometry, there are dozens of identities to learn. I’ve taught the course 20 times, and with my bad memory, I still don’t have all the identities in my head. All the basics are there, but some of the identities I need for Calc II are not in my memory.
For example, it turns out that we need an identity that will change the form of sin2x, in order to evaluate the integral of sin2x. While my students were taking a test on volumes of rotation a few weeks back, I was thinking about integration techniques, which I’d be teaching next. I wanted to solve this problem, and had no table of trig identities handy. So I thought I’d draw a graph to see what it might tell me.
The next day of class I showed it to my students, and today a student who was absent asked to see my 'derivation'. I was tickled. And then I thought maybe some of you might like to see it. This is how I can remember things - visually. Because then everything is connected.
I want to take each y-coordinate on this graph of y=sin x, and square it, to get y= sin2x. The points where y=0 stay put, and so does the point where y=1. The point at (3π/2,-1) moves to (3π/2,1). The portions of the graph very close to y=0 look almost like straight lines, so squaring those portions will get us a shape that’s close to a parabola near the y=0 points.
y = 1/2 - 1/2 * cos(2x), which is usually written as ...
y = (1-cos 2x)/2. (Apologies for my limited html skills.)
Now that you've seen this, can you find the identity for cos2x?